MATH 1130: Show Using an Honest Hilbert Proof that ⊢𝐴∧(𝐴∨𝐵)≡𝐴: Discrete Mathematics 1 Assignment, DC, Canada
|University||Douglas College (DC)|
|Subject||MATH 1130: Discrete Mathematics 1|
Show using an honest Hilbert proof that ⊢𝐴∧(𝐴∨𝐵)≡𝐴.
For this question, your answer must be a Hilbert proof according to Definition 1.4.5, no shortcuts (including meta theorems or absolute theorems from class/book) are allowed.
(a) Prove that for any wffs 𝐴,𝐵,𝐶, D and any propositional variable 𝑝,
(Hint: Using the deduction theorem or Post’s theorem makes this easier, although there are other proofs).
(b) Is it true that 𝐴→(𝐵→𝐶)⊢𝐴→(𝐷[𝑝:=𝐵]→𝐷[𝑝:=𝐶])? If yes, give a proof of this, and if no, find a counterexample.
Consider the string ((((𝑝∨𝑟)≡𝑞)→((𝑟∧(¬𝑝))≡𝑟))→𝑟)
(a) Write a formula-calculation to show this string is a wff.
(b) Is this formula a tautology? Explain why or why not.
(a) Give an example of a well-formed formula 𝐴 so that we do NOT have ⊢¬𝐴. Explain why your answer works (using soundness might help).
(b) What is wrong with the following equational proof of ⊢¬𝐴?
⇔⇔ (¬-introduction axiom)
⇔⇔ (Leibniz from prev. line, “C-part” 𝑝≡⊥)
⇔⇔ (⊤ vs. ⊥)
Use the method we discussed from the “Weak Post’s Theorem” notes to show:
Write an equational-style proof for the following: 𝑟→(𝑝∨𝑞)⊢(𝑟∨(𝑝∨𝑞))→(𝑝∨𝑞).
Give proof to show
(Use equational or Hilbert proof styles. Post’s theorem is not allowed).
What is wrong with the following equational proof of ⊢(𝐴→𝐶)≡𝐴?
⇔⇔ (implication axiom)
⇔⇔ (Leibniz+redundant true; “C-part” 𝐴∨𝑝)
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